homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
The generalization of the bicategory Span to (∞,n)-categories:
An -category of correspondences in ∞-groupoid is an (∞,n)-category whose
objects are ∞-groupoids;
morphisms are correspondences
in ∞Grpd
2-morphisms are correspondences of correspondences
(where the triangular sub-diagrams are filled with 2-morphisms in ∞Grpd which we do not display here)
and so on up to n-morphisms
-morphisms are equivalences of order of higher correspondences.
Using the symmetric monoidal structure on ∞Grpd this becomes a symmetric monoidal (∞,n)-category.
More generally, for some symmetric monoidal (∞,n)-category, there is a symmetric monoidal -category of correspondences over , whose
objects are ∞-groupoids equipped with an (∞,n)-functor ;
morphisms are correspondences in (∞,1)Cat over
and so on.
Even more generally one can allow the ∞-groupoids to be (∞,n)-categories themselves.
The (∞,2)-category of correspondences in ∞Grpd is discussed in some detail in (Dyckerhoff-Kapranov 12, section 10). A sketch of the definition for all was given in (Lurie, page 57). A fully detailed version of this definition is in (Haugseng 14).
In (BenZvi-Nadler 13, remark 1.17) it is observed that
is equivalently the (∞,n)-category of En-algebras and (∞,1)-bimodules between them in the opposite (∞,1)-category of (since every object in a cartesian category is uniquely a coalgebra by its diagonal map).
(This immediately implies that every object in is a self-fully dualizable object.)
To see how this works, consider any object regarded as a coalgebra in via its diagonal map (here). Then a comodule over it is a co-action
and hence is canonically given by just a map .
Then for
two consecutive correspondences, now interpreted as two bi-comodules, their tensor product of comodules over as a coalgebra is the limit over
This is indeed the fiber product
as it should be for the composition of correspondences.
For an (∞,1)-topos and a symmetric monoidal internal (∞,n)-category then there is a symmetric monoidal (∞,n)-category
whose k-morphisms are -fold correspondence in over -fold correspondences in , and whose monoidal structure is given by
This is (Haugseng 14, def. 4.6, corollary 7.5)
If is (or is regarded as) a moduli stack for some kind of bundles forming a linear homotopy type theory over , then the phased tensor product is what is also called the external tensor product.
Examples of phased tensor products include
is a symmetric monoidal (∞,n)-category with duals.
More generally, if is a symmetric monoidal -category with duals, then so is equipped with the phased tensor product of prop. .
In particular every object in these is a fully dualizable object.
This appears as (Lurie, remark 3.2.3). A proof is written down in (Haugseng 14, corollary 6.6).
The canonical -∞-action on induced via prop. by the cobordism hypothesis (see there at the canonical O(n)-action) is trivial.
This statement appears in (Lurie, below remark 3.2.3) without formal proof. For more see (Haugseng 14, remark 9.7).
More generally:
For an (∞,1)-topos, then is an (∞,n)-category with duals.
And generally, for a symmetric monoidal (∞,n)-category internal to , then equipped with the phased tensor product of prop. is an (∞,n)-category with duals
Let be the (∞,n)-category of cobordisms.
The following data are equivalent
Symmetric monoidal -functors
Pairs , where is a topological space and a vector bundle of rank .
This appears as (Lurie, claim 3.2.4).
For references on 1- and 2-categories of spans see at correspondences.
An explicit definition of the (∞,2)-category of spans in ∞Grpd is in section 10 of
An inductive definition of the symmetric monoidal (∞,n)-category of spans of ∞-groupoid over a symmetric monoidal -category is sketched in section 3.2 of
there denoted . Notice the heuristic discussion on page 59.
More detailed discussion is given in
Rune Haugseng, Iterated spans and “classical” topological field theories (arXiv:1409.0837)
Yonatan Harpaz, Ambidexterity and the universality of finite spans (arXiv:1703.09764)
Both articles comment on the relation to Local prequantum field theory.
The generalization to an -category of spans between (∞,n)-categories with duals is discussed on p. 107 and 108.
The extension to the case when the ambient -topos is varied is in
The application of to the construction of FQFTs is further discussed in section 3 of
Discussion of is around remark 1.17 of
A discussion of a version for a 2-category with regarded as a tricategory and then as a 1-object tetracategory is in
A discussion that in a 2-category with weak finite limits is a compact closed 2-category:
See also
Coisotropic orrespondences for derived Poisson stacks:
Last revised on April 26, 2019 at 07:18:48. See the history of this page for a list of all contributions to it.